Revisiting the Siegel upper half plane. I.
This paper consists of two parts. The first deals with the general geometry of the Siegel upper half plane. In particular, the authors consider the compactifications of this space. Their main result is the identity of the compactification as a bounded domain and that obtained using the Busemann function. They also clarify the relationship between this boundary and the Shilov boundary. The second part of the paper is devoted to the study of discrete subgroups of the real symplectic groups, including those of infinite covolume. The authors study the limit sets of these groups and develop the theory of the critical exponents and of the Patterson-Sullivan measures for these groups, both of these with respect to \(p\)-metrics. They also consider various modifications of this concept. They obtain extensive results, for example on the non-vanishing of the critical exponents when the discrete group is ``non-elementary'', i.e. when it is not contained in any strict algebraic subgroup of the symplectic group.
scientific article; zbMATH DE number 2012696
scientific article